In the first part ofthis thesis we consider the cubic Schrödinger equationiu_t+Delta u =+/-|u|^2u, x in T_theta^2,t\in [-T,T],u(x,0)=u_0(x) in H^s(T_theta^2).T is the time of existence of the solutions and T_theta^2 is the irrational torus given by R^2/theta_1Z * \theta_2 Z for theta_1, theta_2 > 0 and theta_1/theta_2 irrational. Our main result is an improvement of the Strichartz estimates on irrational tori using a counting argument by Huxley [43], which estimates the number of lattice points on ellipsoids. With this Strichartz estimate, we obtain a local well-posedness result in H^s for s>131/416. We also use energy type estimates to control the H^s norm of the solution and obtain improved growth bounds for higher order Sobolev norms.In the second and the third parts of this thesis, we study the Cauchy problem for the 1d periodic fractional Schrödinger equation:iu_t+(-Delta)^alpha u =+/- |u|^2u,x in T, t in R,u(x,0)=u_0(x) in H^s(T), where alpha in (1/2,1). First, we prove a Strichartz type estimate for this equation. Using the arguments from Chapter 3, this estimate implies local well-posedness in H^s for s>(1-alpha)/2. However, we prove local well-posedness using direct X^(s,b) estimates. In addition, we show the existence of global-in-time infinite energy solutions. We also show that the nonlinear evolution of the equation is smoother than the initial data. As an important consequence of this smoothing estimate, we prove that there is global well-posedness in H^s for s>(10*alpha+1)/(12).Finally, for the fractional Schrödinger equation, we define an invariant probability measure mu on H^s for s0 there is a set Omega, a subset of H^s, such that mu(Omega^c)2/3.
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